No conflict. The bodies of flesh of those who are alive at the Resurrection die as the saints take on their new "glorified" bodies.
You seem to be suggesting a mass death event to bring these two passages into agreement? No such teaching is found and is not supported by any other passage. The conflict exists when you have one passage suggesting a "rapture" of living people (1 Thes 4:17) vs. another passage which insists everyone must die first (Heb 9:27).
So again, I believe the appropriate way to view Heb 9:27 and 1 Thes 4:17 is to see that those who outlived the resurrection, or were not yet born, are caught up with the resurrected after they each die. This would mean no mass rapture of living and no escaping death. There is no support for the "Fly Away Doctrine." Let's not forget that we have 2 Cor 12:1-2 which discusses the rapture of 1 individual where the same word, Harpazo, is used. This supports my view.